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/Part 3: Functional Analysis/Chapter 11: Locally Convex Spaces/Section 11.10: Locally Convex Spaces of Linear Maps

Lemma 11.10.3.label Let $E$, $F$ be locally convex spaces over $K \in \RC$, $\sigma \subset 2^{E}$ be an ideal, and $\ol \sigma$ be its saturated hull, then the $\sigma$-uniformity and $\ol \sigma$-uniformity on $L(E; F)$ coincide.

Proof. Let $\tau \subset \ol \sigma$ be the collection of sets such that for each $S \in \tau$ and $U \in \cn_{F}(0)$,

\[N(S, U) = \bracs{(S, T) \in L(E; F)| (S - T)(S) \subset U}\]

is an entourage in the $\sigma$-uniformity.

For each $S \in \tau$, $U \in \cn_{F}(0)$, and $\lambda \in K$ with $\lambda \ne 0$,

\begin{align*}N(\lambda S, U)&= \bracs{(S, T) \in L(E; F)| (S - T)(\lambda S) \subset U}\\&= \bracs{(S, T) \in L(E; F)| (S - T)(S) \subset \lambda^{-1}U}\end{align*}

is another entourage in the $\sigma$-uniformity. If $\lambda = 0$, then $N(\lambda S, U) = L(E; F)$, which is also an entourage.

Now, let $S \in \tau$ and $U \in \cn_{F}(0)$ be convex and circled, then by Proposition 5.5.3,

\begin{align*}N(\ol{\aconv}(S), U)&= \bracs{(S, T) \in L(E; F)| (S - T)(\ol{\aconv}(S)) \subset U}\\&\supset \bracs{(S, T) \in L(E; F)| \overline{(S - T)(\aconv(S))} \subset U}\\&= \bracs{(S, T) \in L(E; F)| \ol{\aconv}{(S - T)(S)} \subset U}\end{align*}

so $N(\ol{\aconv}(S), U)$ contains an entourage in the $\sigma$-uniformity.

Since $\tau$ is a saturated ideal that contains $\sigma$, $\tau = \ol \sigma$. Therefore the $\sigma$-uniformity and $\ol \sigma$-uniformity on $L(E; F)$ coincide.$\square$

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